Hull, bounding rectangle, and enclosing circle

Model: Circle: Welzl's smallest circle on an azimuthal equidistant plane, re-centered on its result until the circle's center is the plane's own; that plane keeps distances and directions from its center, so the fixed point is the smallest geodesic circle (Karney 2013). Hull: great-circle hull of the points on the sphere, by monotone chain in a gnomonic projection from their mean. Rectangle: the smallest-area rectangle on an edge of the hull, on the equidistant plane at the points' center

Accuracy: The circle's radius is an exact geodesic distance, the center converged to 1 mm; the rectangle is planar on the equidistant map, true for spans of tens of kilometers to about 1 part in 10⁶

When to use this: Use this to put a shape around a set of positions: the coverage circle for a set of sightings, the smallest area holding a survey's control points, the footprint of a swarm or a fleet, the block a site occupies. Three answers come back because they suit different jobs — a circle is what a range or a broadcast covers, the convex hull is the tightest area that holds everything, and the smallest rotated rectangle is how a field, a runway or a site plan is usually described.

Limitations: All three are computed on one plane placed at the points, so they are meant for spreads of tens of kilometres rather than continental ones; the circle's radius is then an exact geodesic distance while the rectangle stays planar. The convex hull holds every point and says nothing about how they are distributed inside it: one outlier stretches all three answers, and none is a summary of where the points mostly are. Collinear or nearly collinear points give a hull and a rectangle that degenerate to a line, with zero area and zero width, which is correct. Where the hull is a triangle the smallest rectangle is not unique — all three edge-flush rectangles have exactly the same area — so the sides returned are one valid choice among equals.

Worked example: Seven survey points. Source: checked against GEOS 3.11.4 through shapely — its own minimum_bounding_circle, convex_hull and minimum_rotated_rectangle — over twelve point sets, agreeing on every hull corner count, on the circle radius to 2.5e-8 relative, on the hull area to 5.7e-7, and placing the circle's centre within 0.8 m over a 20 km scatter.

Review: Not yet independently reviewed by a GIS professional.

Last verified: 2026-09-18, when a maintainer last confirmed this tool's sources at the issuer. See the sources ledger.

Status: version 1.0.0, core 0.1.0. See this tool in the verification report.

Checked against: 21 golden test vectors (download the test vectors, each with its source and tolerance). See how results are checked and every source.

Sources

• Algorithms for geodesics, Karney, C. F. F., Journal of Geodesy, Vol. 87, No. 1. pp. 43-55 (direct and inverse problems; the vertex of a geodesic, where its azimuth is 90°).